# Questions tagged [numerics]

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### Why is the central difference method dispersing my solution?

I am solving numerically the ODE $\ddot x(t)=-c\dot x(t) -\sin(x(t))+F\cdot \cos(\omega t), \;\dot x(0)=x(0)=0$ for $t\in [0,20\pi]$ on an $N=2000$ dimensional grid. I am working on Python, and I ...
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### "Optimal" domain partitioning in domain decomposition algorithms

When solving a PDE numerically by domain decomposition methods, what is the "optimal way" to split the domain? Are there any results stating that a particular partition of the domain yields &...
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### How is FEM used in structural engineering?

I have learned about the finite element method (FEM) as a method for solving boundary problems given by a PDE. The way I learned it is to approximate the solution by a linear combination of test ...
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### Weird behavior in for solving TISE in harmonic oscillator potential using the shooting method

I was solving the time independent Schrödinger equation using the shooting method for harmonic oscillator potential. This is the code that I wrote for that with the results (code is written in julia): ...
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### How to solve a second order differential equation (diffusion) with boundary conditions using Python

I am having trouble implementing a model from a publication. Huang, K-L.; Holsen, T.M.; Selman, J.R. Ind. Eng. Chem. Res. 2003, 42, 15, 3620–3625 scihub link: https://sci-hub.se/10.1021/ie030109q I ...
732 views

### Error in Simpson's 3/8 rule is higher than that of Simpson's 1/3 rule

For a given function $f(x)$, I have tried to find its numerical integral using Simpson's 1/3 and Simpson's 3/8 rules. I then compare the solution from the numerical quadratures to the analytical ...
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### SIPG method for $-\nabla \cdot (\nu \nabla u)=f$

Consider the diffusion equation with a coefficient $\nu$: $$-\nabla \cdot (\nu \nabla u)=f$$ with Dirichlet boundary conditions $u = g_D$ in $\partial \Omega$. If the coefficient would be constant, ...
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### Typo in a-priori error estimate in a Discontinuous Galerkin paper

I'm looking at this famous paper which is available in the link below: Franco Brezzi, LD Marini, Endre Süli, Discontinuous Galerkin methods for first-order hyperbolic problems, Mathematical Models ...
2k views

### Why do we usually not want the eigenvalues of non-symmetric matrices?

I came across this line in a class note I am reading where it discusses finding eigenvalues of matrices. In reality we don't go all the way with Arnoldi. We stop at a decent value of 𝑘. Then the 𝑘 ...
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### Is there a graphical interpretation or explanation of automatic differentiation compared to numerical differentiation

I have been looking at automatic differentiation for solving differential equations lately. I understand the basic ideas of using Dual numbers and such for finding derivatives, etc. However, I feel ...
162 views

### What is the difference between $u_h$ and $I_h(u)$ in finite element literature?

In finite element books, we have estimates for $$||u-u_h||$$ and also estimates for $$||u - I_h(u)||$$ where $I_h(u): V \mapsto V_h$ projects a function from the infinite dimensional space to the ...
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### Does the time-dependent 1D advection-diffusion with point sources have an analytical solution?

I am looking for the analytical solution of 1-dimensional advection-diffusion equation with several point sources, Q, along the axial length of a cylinder through which the fluid flow occurs. Neumann ...
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### Numerical algorithm to calculate stream of discounted cash flows

Is it possible to calculate the flow of cash in each period given a certain fixed net present value with a numerical algorithm? I would like to be able to apply a certain weighting that can be applied ...
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### How important is learning hardware/architecture for scientific computing?

Apologies if this is a bit of a soft, unclear, or opinion-based question. I'm a relatively new PhD student in a (computational) quantum chemistry group. My group develops and maintains a few software ...
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### How to implement point source or volume source in finite element implementations

I'm trying to do a simple implementation to study the advection-diffusion-reaction dynamics in a straight pipe. I have points positioned along the length of the pipe (blue dots in the image above). I ...
I would like to ask a question on the stability of BDF2 applied to hyperbolic PDEs. Say I have a hyperbolic equation as $\frac{\partial c}{\partial t} + {\bf U} \cdot {\bf \nabla}c=0$. This system is ...